Computers have provided opportunities for significant advances in educational processes. They have been increasingly used in teaching, particularly for young children who have become extremely adept at using computers. Much educational software is based upon a teacher or tutor model, in which the system guides the user in specific areas with predefined objectives and information to be learned. Such systems interpret different levels of achievement and assume that if the information given is below the user's level, the user may become bored; if the information is above the user's level, the user may become frustrated. Also, under a strict teaching approach, the user has no control over what is being learned. Points of interest to a specific user may be missed. Often instructional software, while presenting important information, is not particularly enjoyable to use; nor is the user's resulting understanding particularly memorable. Therefore, a need exists for learning environments that permit a user to learn material at his or her own pace, exploring to depths of interest, and which are entertaining.
Complex, dynamic systems can be difficult to understand. Such systems do not operate according to a single set of rules that can be easily learned. Instead, they operate as dynamic networks of interdependent elements, which vary in unique ways based upon a large variety of factors present at a given point in time. The world is made up of great numbers of such systems, in both biological and physical domains. Examples of such systems are population growth, weather patterns, economic fluctuations, biological evolution, organizational behavior, and traffic patterns. Since it is difficult to describe such complex systems, not many tools are available for exploring and learning about them. Therefore, a need exists for a system which provides opportunities for learning about and understanding how these complex processes operate.
Mathematics, particularly the mathematics of geometry and spatial relationships, provides opportunities for studying complex systems in ways that can be made simple, enjoyable, and understandable. One way for people, particularly children, to develop mathematical and spatial skills is by working with patterns. Thus, educators and parents encourage children to play with toys that use and manipulate patterns, such as tilings, kaleidoscopes, moire patterns, quilts, mosaics, tangrams, and geometric puzzles. The use of such playthings permits a child to learn by increasing his or her spatial skills while engaged in entertaining activities. However, many existing toys minimize control by the user, limit constructive capabilities that are useful for learning, do not allow for lengthy, rigorous explorations, and cannot capture the dynamic aspects of complex systems.